Nonparametric relative error estimation of the regression function for censored data

Abstract

Let $ (T_i)_i$ be a sequence of independent identically distributed (i.i.d.) random variables (r.v.) of interest distributed as $ T$ and $(X_i)_i$ be a corresponding vector of covariates taking values on $ \mathbb{R}^d$. In censorship models the r.v. $T$ is subject to random censoring by another r.v. $C$. In this paper we built a new kernel estimator based on the so-called synthetic data of the mean squared relative error for the regression function. We establish the uniform almost sure convergence with rate over a compact set and its asymptotic normality. The asymptotic variance is explicitly given and as product we give a confidence bands. A simulation study has been conducted to comfort our theoretical results